- defining differential operator (as gradient and divergence) on manifold
- understand how Laplacian is defined thanks to Laplace-Beltrami operator
- understand PDEs on manifold and define the Brownian motion on it
- how to construct the Brownian motion on Riemanian manifold with the Laplace-Beltrami operator
- several concrete examples

- Newtonian Classical Mechanics (affine spaces, Galilean transformations, Newton's equation)
- Newtonian Classical Mechanics (examples, kinetic and potential energy)
- Lagrangian Classical Mechanics (motivation, calculus of variations)
- Lagrangian Classical Mechanics (Lagrangian function and action, Euler-Lagrange formula, Hamilton's equation)
- Observables (algebra of observables, Poisson-bracket, analogies to quantum mechanics)
- Axioms of Quantum Mechanics (states, observables, measurement, correspondence principle, position and momentum operators)
- Schrödinger Equation (wavefunction, Hamilton operator, eigenvalue equation and eigenstates)
- Schrödinger Equation continued (eigenstates for potential well and potential step), Double-Slit Experiment (interference, probability amplitude calculus)

- Tangent Vectors, The Differential of a Smooth Map, The Tangent Bundle (reference: “Introduction to Smooth Manifolds” by John M. Lee)
- Connection, Vector Fields, Covariant Derivative, Geodesic and Parallel Translation
- Discussion about curvature of space and Gauss-Bonnet Theorem
- Divergence theorem, Green's identities
- Laplace–Beltrami operator
- Lie group and principal bundle

- differentiable structures, manifolds, tangent spaces
- de Rham cohomology, embeddings of manifolds, regular value theorem
- (pseudo-)Riemannian manifolds, Riemannian measures, Laplace-Beltrami operator, Gauss-Green identity, Stokes' theorem
- Covariant derivative, Riemannian connection, Christoffel symbols, Parallel displacement, exponential mapping (cf. Section 5D, Kuehnel: Differential geometry)
- Curvature tensor and intrinsic geometry of surfaces: Equations of Gauss and Weingarten, Theorema egregium, Fundamental theorem of the local theory of surfaces (Sections 4A-D, Kuehnel)
- More on the curvature tensor and Weingarten map, Sectional curvature, Ricci tensor and curvature, Einstein tensor (Chapters 4,6: Kuehnel)
- Spaces of constant curvature: Hyperbolic space, Geodesics and Jacobi fields, Local isometry of spaces of constant curvature (Chapter 7: Kuehnel)

- Organizational meeting
- Probability that random polynomial has no real roots (Talk by Anna Gusakova)
- Theorems on positive real functions (Talk by Anna Muranova)
- Non-explosion of solutions to SDEs with singular coefficients (Talk by Chengcheng Ling)
- Emergence of Flocking in the Fractional Cucker-Smale Model (Talk by Peter Kuchling)
- Robust Poincaré inequality and Application to the transitional phase of fractional Laplacian (Talk by Guy Fabrice Foghem Gounoue)
- My problems and failed ideas regarding the edge fluctuation of non hermitian random matrices with independent entries (Jonas Jalowy)
- Effective impedance of a finite electric network. Definitions and main properties (theorems and conjectures) (Anna Muranova)
- Application of Probability Methods in Number Theory and Integral Geometry (Anna Gusakova)
- Ordered fields. The ordered field of rational functions“ (Anna Muranova)
- Dynamics on the cone: an overview of my thesis (Peter Kuchling)
- Well-posedness of SDE driven by Levy noise (Chengcheng Ling)
- Discussion about the talks for workshop/retreat

- Point processes 3: Moment Problem, Local Convergence Jansen, Gibbsian Point Processes, pp. 33 - 45
- Point processes 2: Intensity Measure, Correlation Functions, Generating Functionals Jansen, Gibbsian Point Processes, pp. 25 - 33
- Point processes 1: Configuration Spaces, Observables, PPP Jansen, Gibbsian Point Processes, pp. 15 - 25
- Introduction to partition functions
- Partition Functions and Gibbs Measures
- Correlation Functions
- Phase Transitions (Critical Exponents and Models)
- Scaling Limits
- Mean Field Limit of the Cucker-Smale model: Two Approaches (Peter Kuchling)

**Reading** *Spectral graph theory* by Fan R. K. Chung

- Chapter 2: Isoperimetric problems
- Chapter 3: Diameters and eigenvalues
- Talk by Anna Muranova: On effective resistance of electric network with impedances
- Talk by Melissa Meinert: Ollivier Ricci curvature for general Laplacians
- Markov chains
- Talk by Filip Bosnic: Examples of Poincaré and log-Sobolev inequalities on discrete configuration spaces
- Introduction to random graphs by Alan Frieze and Michał Karoński

**Discuss** *Lévy Processes and Infinitely Divisible Distributions* by Ken-iti Sato

- Chapter 1 (pp. 1 - 30): Examples of Levy Processes

- Chapter 2 (pp. 31 - 68): Characterisation and Existence of Levy Processes and Additive Processes
- Chapter 2 (pp. 31 - 41): Infinitely Divisible Processes and the Lévy-Khintchine formula
- Chapter 2 (pp. 54 - 67): Transition Functions and the Markov Property; Existence of Lévy and Additive Processes
- Chapter 3 (pp. 69 - 77): Selfsimilar and Semi-selfsimilar Processes and their Exponents
- Chapter 3 (pp. 77 - 90): Representations of Stable and Semi-stable Distributions
- Viswanathan et al. Papers: “Levy Flight Search Patterns of Wandering Albatrosses” and “Optimizing the Success of Random Searches”
- Bertoin (pp. 18-24): Potential theory: Markov property and important definitions
- Bertoin (pp. 43-48): Duality and time reversal
- Bertoin (pp. 48-56): Potential theory: capacity measure; essentially polar sets and capacity
- Bertoin (pp. 56-68): Potential theory: energy; the case of a single point
- Bertoin (pp. 71-84): Subordinators: definitions; passage across a level; arcsine laws
- Bertoin (pp. 84-99): Subordinators: rates of growth; (Hausdorff-)dimension of the range

**Discuss**

- Maximum principle for ecliptic operators (
*Gibarg-Trudinger*) - Paley-Littlewood Decomposition (
*Grafakos classical Fourrier Analysis, second or third edtion*) - Another looks at Sobolev spaces (
*Articles Brezis Bourgain_Mironescu*) - Spectral theory: spectral measure for Unbounded operators
- Spectral theory for unbounded operator and spectral measure (Book by Reed-Simon)
- Restriction of the Fourier Transform on the sphere (Book by Elias Stein)
- Interplation of Banach spaces and application (Lecture note by Alessandra Lunadri Theory of interpolation 2009)
- Introduction to optimal transport theory (Book by Cedri Villani: Old and New, it is free on-line)
- Introduction to control and non linearity (Book by Jean-Michel Coron: Control and Non-linearity free one his we-page)
- Introduction to Calculus of variations